Modelling and Optimisation of a Free Piston Stirling Engine for Micro-CHP Applications

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Citation: Sowale, Ayodeji (2015) Modelling and Optimisation of a Free Piston Stirling Engine for Micro-CHP Applications. Doctoral thesis, Northumbria University.

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Modelling and Optimisation of a Free Piston

Stirling Engine for Micro-CHP Applications

Ayodeji Olubayo Sowale

A thesis submitted in partial fulfilment of the requirements

of the University of Northumbria at Newcastle for the award

of Doctor of Philosophy

Research undertaken in the Mechanical and Construction

Engineering Department of the

Faculty of Engineering and Environment

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Abstract

This study is carried out to investigate the solar thermal energy conversion for generating power. This form of renewable energy can be utilised for power production deploying the free piston Stirling engines, which convert thermal energy into mechanical energy. Such systems have an advantage of production of work using low and high temperature differences in the cycle which could be created by different sources of heat including solar energy, combustion of a fuel, geothermal energy, nuclear energy or waste heat. The thermodynamic analysis of the free piston Stirling engine have been carried out and implemented in past studies with different methods of approach with various difficulties exhibited. In the present study isothermal, ideal adiabatic and Quasi steady flow models have been produced and used for investigation of the engine performance. The approach in this study deals with simultaneous mathematical modelling of thermodynamic processes and pistons dynamics. The steady state operation of the engine depends on the values of damping coefficients, spring stiffness and pressure drop within the heat exchangers during engine’s operation, which is also a result of the energy transfer in each engine’s component. In order to design effective high performance engines it is necessary to develop such advanced mathematical models to perform the analysis of the engine’s operation and to predict its performance satisfactorily. The aim of this study was to develop several levels of mathematical models of free piston Stirling engines and to evaluate their accuracy using experimental and theoretical results available in published sources. The validation of the developed free piston Stirling engine models demonstrates a good agreement between the numerical results and experimental data.

The validated model then was used for optimisation of the engine, deploying Genetic Algorithm approach with the purpose to determine its optimal design parameters. The

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developed optimisation procedure provides a noticeable improvement in the engine’s performance in terms of power output and efficiency.

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List of Figures

Figure 1.1: A Schematic diagram of a CCHP/CHP system with Stirling engine ... 2

Figure 1.2: P-V diagram of the Stirling cycle. ... 3

Figure 1.3: The simplified schematic diagram of the Stirling engine. ... 4

Figure 2.1: Schematic diagram of a dual free piston Stirling engine ... 14

Figure 2.2: Notation of a typical beta type free piston Stirling engine ... 15

Figure 2.3: Layout diagram of a free piston Stirling engine. ... 19

Figure 2.4: Sunpower B-10B free-piston Stirling engine demonstrator. ... 20

Figure 2.5: Geometry of the free piston Stirling engine. ... 20

Figure 2.6: Cut away view of RE-1000 free-piston Stirling engine ... 25

Figure 2.7: Sunpower EG-1000 free piston Stirling engine ... 32

Figure 2.8: Sunpower EE-35 free piston Stirling engine ... 33

Figure 3.1: A schematic diagram of the alpha type Stirling engine ... 36

Figure 3.2: The Ross yoke engine ... 36

Figure 3.3: A multi cylinder configuration of the alpha type Stirling engine. ... 37

Figure 3.4: A schematic diagram of the beta type Stirling engine ... 38

Figure 3.5: A schematic diagram of the gamma type Stirling engine ... 39

Figure 3.6: A schematic diagram of the Stirling engine isothermal model with temperature distribution. ... 46

Figure 3.7: Variation of volumes in a Stirling engine. ... 46

Figure 3.8: A schematic diagram of the gamma type free piston Stirling engine ... 58

Figure 3.9: The Air Independent Propulsion system ... 61

Figure 3.10: The graph of Torque-Crankshaft Angle ... 62

Figure 3.11: The graph of Altitude-Airspeed ... 63

Figure 3.12: Conceptual design of the SRG by Lockheed ... 64

Figure 3.13: A schematic diagram of the solar dynamic Brayton ... 65

Figure 3.14: A domestic Micro-CHP system ... 66

Figure 4.1: The temperature profile of the isothermal model ... 69

Figure 4.2: The layout diagram for the dynamic analysis of the free piston Stirling engine used in this study. ... 70

Figure 4.3: The linear profile temperature of the regenerator ... 72

Figure 4.4: Variations in expansion and compression spaces over the cycle ... 74

Figure 4.5: Working fluid pressure variation and displacement of pistons over the cycle ... 76

Figure 4.6: Gas pressure variations in the expansion and compression spaces ... 79

Figure 4.7: The flow chart of the developed isothermal model of the free piston Stirling engine. .. 83

Figure 4.8: The graph of dimensionless cyclic work against beta for beta FPSE ... 84

Figure 4.9 The graph of dimensionless cyclic work against beta for gamma FPSE. ... 85

Figure 5.1: The Ideal Adiabatic model ... 89

Figure 5.2 The flow chart of the simulation in the accordance with the developed second order of the adiabatic model of the free piston Stirling engine. ... 98

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Figure 5.4: The graph of gas temperature variation in the expansion and compression space

variation. ... 100

Figure 5.5: The graph of displacer velocity against displacer amplitude at steady state. ... 100

Figure 5.6: The graph of piston velocity against piston amplitude at steady state. ... 101

Figure 5.7: The graph of expansion and compression space volume variation. ... 102

Figure 5.8: The graph of pressure - volume diagram of the engine. ... 102

Figure 6.1: The layout diagram of the Re-1000 free piston Stirling engine. ... 105

Figure 6.2: The control volumes of the engine ... 106

Figure 6.3: Temperature distribution in the work spaces and heat exchangers. ... 106

Figure 6.4: The flow chart of calculations using the developed second order Quasi steady flow model of the free piston Stirling engine. ... 119

Figure 6.5: The Pressure - volume diagrams for expansion and compression spaces ... 120

Figure 6.6: Heat flow rate in the heat exchangers. ... 121

Figure 6.7: Volume variation in expansion and compression space ... 122

Figure 6.8: The displacement of piston at a frequency of 30Hz ... 122

Figure 6.9: The displacement of displacer at an operating frequency of 30Hz. ... 123

Figure 6.10: The velocity of the piston and displacer. ... 124

Figure 6.11: Mass flow rates of the working fluid in the heat exchangers. ... 125

Figure 6.12 Pressure –total volume diagram ... 126

Figure 6.13: Temperature variations in the heat exchangers, the first and tenth part of the regenerator matrix. ... 126

Figure 6.14: Temperature variation in the expansion and compression spaces. ... 127

Figure 6.15: Reynolds number variation in the chambers of the engine... 128

Figure 6.16: The bounce space pressure (buffer pressrue) in the piston compartment. ... 129

Figure 6.17: The mass of the working fluid in the heat exchangers and work spaces. ... 129

Figure 6.18: Internal heat conduction loss in parts of the regenerator. ... 130

Figure 6.19: External heat conduction loss in parts of the regenerator. ... 131

Figure 6.20: The pressure drop in the heat exchangers. ... 131

Figure 6.21 Relative positions of the piston and displacer in the engine during operation. ... 132

Figure 7.1: Flow chart of Genetic algorithm. ... 137

Figure 8.1 Comparison of the pressure variation in FPSE obtained using the developed Quasi steady flow and adiabatic models ... 145

Figure 8.2: Comparison of the temperature variation in expansion and compression spaces obtained using the developed Quasi steady flow and adiabatic models. ... 146

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List of Tables

Table 3.1: Relative heat transfer characteristics of various gases [99] ... 48

Table 3.2: Relative performance of selected working fluid [99] ... 48

Table 3.3: Parameters of Stirling engine CHP systems [109] ... 67

Table 4.1: Parametric values used for the numerical modelling of the FPSE ... 81

Table 4.2: Comparison of the outputs from isothermal model of the and FPSE. ... 85

Table 5.1: Engine data of the RE-1000 free piston Stirling [97] ... 97

Table 5.2 Comparison of the Sunpower RE-1000 FPSE results with numerical results from the developed model. ... 103

Table 6.1: The effects of mean pressure on the engine performance ... 133

Table 6.2:The effects of damping in the engine ... 133

Table 6.3: The effects of heater temperature in the engine. ... 133

Table 8.1: Comparison of the experimental and theoretical results on the FPSE power production ... 144

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Nomenclature

A = cross sectional area (m2)

� = Cross sectional area of the piston ( ) � = Cross sectional area of the displacer ( )

= specific heat at constant pressure (J/ kgK)

= specific heat at constant volume (J/ kgK)

d = diameter (m)

diss = heat loss due to the flow friction in the regenerator

FPSE = Free piston Stirling engine

= Stiffness of piston spring / ) = Stiffness of displacer spring ( / ) = thermal conductivity

= length

lir = heat loss due to heat transfer due to the heat conduction

= Mass of the displacer = Mass of the piston

Npop = number of chromosome

NTU = number of heat transfer unit

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xi = number of survival chromosomes

= Bounce space pressure =chromosomes

P = power (W)

̅ = mean gas pressure in the cycle

Pr = Prandtl number

Q = heat transfer rate (W)

= Temperature of the cooler (K) = Temperature of the regenerator (K) ℎ = Temperature of the heater (K)

= Temperature of the compression space to cooler (K) ℎ = Temperature of the heater to expansion space (K)

ℎ = Temperature of the regenerator to heater (K) = Temperature of the cooler to regenerator (K) = Bounce space volume )

= Swept volume of the compression space ) = Swept volume of the expansion space ) = Volume of the regenerator )

ℎ = Volume of the heater ) = Volume of the cooler )

= Ratio of heater volume to expansion space swept volume = Ratio of cooler volume to compression space swept volume

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= Ratio of regenerator volume to expansion space swept volume = Ratio of bounce space volume to expansion space swept volume = Ratio of bounce space volume to compression space swept volume

= variables

= Compression space clearance volume ) = Expansion space clearance volume )

V1 = is maximum of the expansion volume. )

W = work done (J) = Displacement of piston ( ) = Velocity of piston ( / ) = Acceleration of piston / = Displacement of displacer ( ) = Velocity of displacer ( / ) = Acceleration of displacer /

R = gas constant value

Re = Reynolds number

= selection rate

ℎ = shuttle heat loss from the heat transfer to the cooler from the heater p = pressure drop (Pa)

Subscripts

c = compression space d = dead volume

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e = expansion space

freesurf = free surface area

g = working fluid

ℎ = heater

in = indicated

= parts of the regenerator matrix = cooler = regenerator matrix = maximum = number = regenerator = value of chromosome Greek Symbols

= angle in Schmidt analysis = is emissivity

θ = phase angle

α = difference in compression space and expansion space phase = density (kg/ )

μ = dynamic viscosity = pressure coefficient

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Declaration

I declare that the work contained is this Thesis has not been submitted for any other award and it is all my own work. I also confirm that this work fully acknowledges the ideas, opinions and contributions from the work of others.

Word count of main body of Thesis: 35,275

Name: Ayodeji Sowale

Signature:

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Acknowledgement

I wish to express my sincere appreciation to my supervisor Prof. Khamid Mahkamov for his invaluable support and encouragement throughout my study.

I will also like to thank Dr. Kwanchai Kraitong for his assistance. I would also like to thank my friends in Northumbria University for their support.

Finally I will like to give special thanks to my parents Rt. Rev Dr. and Mrs Sowale for their constant care, financial and moral support throughout my study.

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Chapter 1 Introduction

Numerous methods have been researched to create an alternative means of energy saving to benefit the environment. These include the combined heat and power generation on micro scale for domestic applications. The financial and environmental benefits of the micro CHP have given it an advantage as an uprising source of energy for use in residence and medium scale commercial environment. The use of micro CHP will decrease the consumption of primary energy, emission of CO2 and the end user’s bill. Based on research and investigations results, Stirling engines are considered favourably for micro CHP generation due to their ability to use various fuels, low level of harmful emissions, good performance at partial load, relatively high efficiency and quietness of operation [1]. In the energy market today, there is an increase in demand for systems that can produce savings in terms of fuel and CO2 emissions [2]. This is a major reason for a strong interest in the use of cogeneration technologies. To reduce the negative environmental effect and increase the efficiency of energy conversion, a number of effective decentralized energy systems are being developed [3]. Stirling engines have been tried for use in different applications since its invention by Robert Stirling in 1816 [4]. Prototype Stirling engines have been designed and tested in transport industry ( such as buses, trucks and boats) [5]. They have been also used as a propulsion engine in passenger ships [6]. Most recent application of Stirling engines are for Micro-CHP with high overall efficiency and the ability to use various fuels.

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2 Absorption Chiller Stirling Engine Heat exchanger Power Heating Cooling Fuel or Heat CHP System CCHP System

Figure 1.1: A Schematic diagram of a CCHP/CHP system with Stirling engine [7].

Certain features of the Stirling engines, which make these an attractive technology for conversion of energy, are as follows:

(i) A very low level of polluted gases emitted from the exhaust to the environment. (ii) The operation of the engine is free of vibration and quiet.

(iii) A very low level of fuel consumption.

(iv) Engines are capable of using different fuels as the source of energy along with solar, geothermal and nuclear energy.

(v) The engine has a reversible mode of operation.

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1.1 The ideal Stirling cycle

Figure 1.2: P-V diagram of the ideal Stirling cycle. [8]

The ideal Stirling cycle, shown in Figure 1.2, consists of two isochoric and two isothermal processes. In the isothermal processes that occur in AB and CD, the working fluid exchanges heat with the heat source and sink and is maintained at constant temperatures. Whilst in the two isochoric processes that occur in BC and DA, there is an internal exchange of heat transfer between the heat exchangers and working fluid, resulting in the rise or reduction in the temperature of the working fluid. In the engines with drive mechanisms the volume changes are predefined. To model and optimize the free piston Stirling engine is not an easy task because of no mechanical linkage between the piston and displacer, therefore it is difficult to determine their strokes and phase angles. To provide the stable operation of the engine, its design parameters are defined using a simultaneous analysis of the working process and dynamics of moving elements is necessary. The isothermal analysis of the working process is usually employed at initial stages of modelling. In isothermal analysis the dynamic pressure can be obtained as a

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function of the positions of the displacer and piston. Hence, the engine performance is obtained deploying linearization methods [9-12]. This analysis does not take into consideration the thermal losses and the effect of heat exchangers on the engine thus leading to an imperfect prediction of the engine performance.

It is required to develop advanced mathematical models to carry out the analysis, taking into consideration the thermodynamic properties of the whole system. In order to predict the accurate performance of the free piston Stirling engine, there is need for application of the second order modelling which takes into account all the pressure and thermal losses within the engine during operation. It still can be achieved by employing thermodynamic isothermal models with the dynamic analysis of the FPSE. A Stirling engine is defined as a mechanical device that produces power through the transfer of heat from a hot temperature region to a cold temperature region [13]. The Figure 1.3 represents the schematic diagram of a simplified Stirling engine for these types of models.

Figure 1.3: The simplified schematic diagram of the Stirling engine [14].

The engine consists of a displacer, a piston, two springs and the engine casing. The displacer and the piston are connected via springs to the casing of the engine. There is no mechanical linkage between the piston and the displacer. Their synchronous motion is achieved by the means of the gas pressure in the working volume of the engine. The

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displacer is required to have a low mass for achieving necessary phase angle between motions of pistons. The inner volume of the engine consists of the dead and working volumes. The working volume is made up of the cold and hot volumes and the dead volume consists of volumes of heat exchangers (cooler, regenerator and heater). Heating, regeneration and cooling processes of the working fluid are achieved in corresponding heat exchangers. The regenerative volume can be seen in the figure above as the gap between the cylinder wall and the displacer’s side surface.

The working cycle of the engine is produced if the hot end of the cylinder is heated up from an external source of energy to a certain degree of temperature, the pressure in the working volume increases in proportion to the ratio of temperature increase and this causes the movement of the piston and displacer downwards. Due to the smaller mass of the displacer it achieves a higher velocity and forces the working fluid from the cold volume to the hot volume. This results in further rise in the pressure of the working fluid which pushes the displacer and piston further downwards. While the displacer is moving from its equilibrium position, the force the displacer spring exerts on it prevails the pressure force resulting in dispacer deceleration. As displacr deceleration starts the working fluid starts to flow in the oppite direction causing the pressure drop and returning of pistons in their intial position.

The thermodynamic analysis of the free piston Stirling engine has been performed in a number of past studies using different methods. This research work aims to extend further the developed advanced modelling and optimization methods for designing free piston Stirling engines for micro-CHP applications. In the present study isothermal, ideal adiabatic and Quasi steady flow models have been developed and used for investigations.

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1.2 Objectives

The main objective is to developed mathematical models required to optimize the design and performance of the free piston Stirling engine operating at steady conditions, by taking into consideration the displacer and piston’s dynamics. The tasks are:

 To study the different methods and approaches to the modelling and optimization of free piston Stirling engines.

 To develop mathematical models with the set of equations to describe working process and dynamics of pistons and to analyse the general characteristics and behaviour of a free piston Stirling engine.

 To develop computer programs to perform the isothermal, adiabatic and Quasi steady flow analysis of the free piston Stirling engine’s operation using MATLAB environment.

 To produce data to present the performance of the engine.

 To compare the outputs generated from the numerical model to real data available in open published sources in order to validate the developed models.

 Perform a parametric check to examine the impact on the performance of the engine model of key working process and design parameters.

 To develop a method for optimization of the free piston Stirling engine.

1.3 Methodology of research

The work in this study was divided into different stages which are as follows:

1. The isothermal model of the free piston Stirling engine was developed for preliminary analysis.

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2. Numerical analysis was carried out to produce feasible values of design parameters of the engine which then could be rectified with deployment of more complex modelling and optimisation methods.

3. The second order mathematical model (based on the adiabatic model) of the free piston Stirling was developed considering the variation in temperatures of the adiabatic working spaces and the gas mass flows of the working fluid during engine’s operation.

4. Numerical analysis was carried out to obtain data on using the adiabatic model and the results were compared with the experimental data for the RE-1000 FPSE, published in open literature.

5. The second order mathematical simulation code using the Quasi steady flow model of the free piston Stirling engine was developed, taking into consideration the thermal losses in the cycle, pressure drop and mass flow rates and the dynamics of piston motions.

6. Numerical simulations were carried out using the Quasi steady model and of losses taking place during the engine’s operation. Theoretical results were compared with the experimental data of the RE-1000 FPSE, published in open literature, and the model was modified.

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7. The developed Quasi steady flow model of the free piston Stirling engine was coupled to a Genetic Algorithm optimization code and optimal design parameters of the engine were obtained.

Thesis structure

The thesis is divided into nine chapters and described below. The structure of the thesis reflects stages of investigations described above.

Chapter 1 Introduction: This chapter is a summary of the aims of the research work performed and describes the structure of this thesis. The contribution to original knowledge was also highlighted in this Chapter.

Chapter 2 Literature review: In this chapter previous studies on the development of the Stirling engine and the free piston Stirling engine were reviewed. Results published from past investigations on the design, experimental tests and numerical simulations of the Stirling and free piston Stirling engines were described. Also different optimization methods used for free piston Stirling engines to obtain the engine’s optimal design were reviewed.

Chapter 3 Theory and applications of Stirling engines: In this chapter the theory behind the operation of Stirling engines is described, their applications, different types of Stirling and free piston Stirling engine configurations and factors governing their performance are discussed.

Chapter 4 General principles and isothermal mathematical modelling of the free piston Stirling engine: In this chapter, the principles of the isothermal model for the free piston Stirling engine are discussed. The mathematical model is defined with all the required

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equations employed to carry out the numerical simulation. The results generated from the numerical simulations were presented and analysed.

Chapter 5 General principles and mathematical modelling using the adiabatic model of the free piston Stirling engine: In this chapter the principles of the adiabatic model of the free piston Stirling engine are discussed with the mathematical equations presented for the second order modelling of the engine. The results generated were presented and analysed.

Chapter 6 General principles of the second order mathematical modelling using the Quasi steady flow model of the free piston Stirling engine: The principles of the Quasi steady flow model of the free piston Stirling engine are discussed in this chapter. The mathematical equations employed to carry out the numerical simulation of the engine were presented and the results obtained from numerical simulations were analysed.

Chapter 7 The principles of Genetic Algorithm: This chapter presents the general principles of the Genetic Algorithm optimization procedure which was employed for the optimization based on the Quasi steady flow model of the free piston Stirling engine.

Chapter 8 Mathematical modelling and optimization of the free piston Stirling engine based on the Quasi steady flow model: This chapter presents the outcome of simulations using the Genetic Algorithm code employed to obtain the optimal engine design parameters. The optimal design parameters were found and the improvement in the output results was demonstrated.

Chapter 9 Conclusions and recommendations for future work: In this chapter the conclusions drawn from the research and investigations in this study are presented and recommendation for future work is made.

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1.5 Original contribution to knowledge

Stirling engines, especially in free-piston configurations, have advantages which make them attractive for micro-CHP applications but currently the technology has a limited use because it is yet to be fully developed. To advance the technology, accurate mathematical methods of its modelling for design purposes should be developed. Due to difficulties in description of the engine’s operation, relatively simple models are in use, which simplify the description of processes taking place in the engine. One of the major simplifications is that the analysis of the working process of the engine and of dynamics of pistons is separated. This results in the low accuracy in determination of required design parameters and the engine’s performance.

The main contribution of this work is that approach has been developed which allows simultaneous modelling of the working process of free-piston Stirling engines and of pistons dynamics. This is accomplished for models of different levels: the first order, advanced second order and Quasi steady flow model of the free piston Stirling engine. Results of the low level models are then used for modelling at the following stage with deployment of the model of higher level. Finally, the optimisation procedure was developed, based on application of the Quasi steady flow model coupled to Genetic Algorithm approach, which provides determination of optimal design parameters of the engine to achieve the specified power output.

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Chapter 2 Literature Review

2.1 Introduction

Previous works related to the development and modelling of Stirling and free Piston Stirling engines are critically reviewed in this chapter. The chapter presents published results on the experiments, designing and simulations of Stirling and free piston Stirling engines.

2.2 Stirling engine modelling

In this section the past studies on the mathematical modelling of the zero order, first, second and third orders of Stirling engines and free piston Stirling engines are presented. Also the use of Genetic Algorithm methods to carry out optimisation of Stirling engines and of other optimisation techniques is described. The fundamentals of the working principles of the free piston Stirling engines are described. Stirling engine simulation and modelling was traditionally performed in the sequence using zero, first and second order models, with each step leading to a more accurate approximation of the engine’s operation in the reality.

2.2.1 Zero order modelling

The zero order modelling is an empirical approach to modelling the Stirling engines rather than the mathematical approach. William Beale introduced the zero order modelling of the Stirling engine as a result of processing a large number of statistical experimental data [15]. Its principle requires more experimental observation other than the special

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on the analysis of the engine cycle with focus on the polytropic process in the power piston and displacer volumes [16]. A model was developed by Fiedt et al on the general law of heat transfer at the heat source and heat sink based on the ideal cycle [17]. Timonmi et al performed a research on the Stirling engine model with focus on the incomplete

regeneration and irreversibility on the ideal cycle [18].

2.2.2 First order modelling

This is an analytical approach to the study of ideal Stirling cycle. The first order model produces performance outputs of Stirling engines overestimated than in the reality due to us of assumptions to idealise the cycle. Gustav Schmidt produced the first order modelling of the ideal Stirling cycle [19]. The result of the analysis produces information on outputs with a low accuracy, but the approach is generally preferred over the zero order. Though it relies on the assumptions of ideal parameter values, it has proven to be a very productive tool for analysing and predicting the engine cycle to generate realistic performance outputs for the engine. Finkelstein conducted a research on the isothermal model of the engine where he tried to proffer solution to the problems encountered in the Schmidt analysis [20]. Kongratool and Wongwise generated an isothermal model of the engine and established the significance of defective regeneration and dead volumes of the system [21]. A related model was developed by Walker on the engine’s four dimensional geometry [13].

The method of approach requires removal of the components of the engine design geometry into a separate subordinate design phase. It proved as an advantage in the engine design but its validity was not generally accepted. [4]. Schmidt’s model is not sufficient for use as the major approach for the Stirling engine design because it uses ideal assumptions to predict performance of the engine than in reality. Schmidt proposed a close form

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expression for the cyclic work of the Stirling engine by developing a mathematical model based on the mass conservation equation. The analysis predicted the behaviour and characteristics of the Stirling engine [22]. The Carnot efficiency is achieved with the Schmidt model due to made assumptions. The model has shown the possibility to align the ideal model to the performance of the actual engine by employing certain correlation factors. The Schmidt analysis was conducted on the Stirling engine using the dead volume with assumptions for isothermal conditions and ideal regeneration [22]. The isothermal and non-isothermal model of the FPSE was conducted by Ulusoy in order to analyse the nonlinear effects. The effects that were studied are non-linear pressure loss, non-linear damper load and gas spring of the displacer, the coefficient of non-linear load terms was used to obtain the stable operation of the periodic motions for the piston and displacer. The non-isothermal behaviour of the working fluid in the FPSE during operation resulted in the difference in temperature. From the analysis a conclusion was made that an isothermal analysis is suitable and more predictive than the non-isothermal analysis for a qualitative dynamic analysis of the engine because the temperature of the working fluid had no noticeable effect on the engine dynamics [23]. There have been a number of approaches to modify and improve the performance of the Stirling engine. The changes have been applied to Stirling engines over the years.

Heat pipes and capillary pump loops were used by Kroliczek et al to produce a more effective means of heat transfer[24]. Also research was performed by Abdulrahman on the engine regenerator as to design and carry out tests on the material to be used to make it more effective [25]. Also various types of working fluids and materials for displacer and piston have been investigated to find the most suitable ones for better engine performance. Although the Stirling engine might not be efficient enough for applications with expected large power output, their mode of operation can be employed in wider applications. The

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general conditions of the practical behaviour of the Stirling engine like the adiabatic internal heat exchangers and heat transfer in the surrounding of the Stirling engine have not been critically examined but the ideal thermodynamics have been analysed. The factors that affect the engine operation in reality need to be considered and examined in order to discover the influence on the actual engine. The temperature of the working fluid in the engine during operation is more likely to behave in an adiabatic manner than isothermal. Thermodynamic evaluation of irreversible Stirling and Ericsson cycles has been performed in [26]. The analysis was based on the perfect and ideal gas regeneration for losses in heat sink with engine power output. The performance of Stirling engine is determined by the various conditions of heat transfer. The research conducted by Costea and Feidt showed that the temperature difference of the engine’s hot components varies linearly with the heat transfer coefficient [27].

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Figure 2.2: Notation of a typical beta type free piston Stirling engine [9].

2.2.3 Second order modelling

The second order modelling of the Stirling engine is a further development of the first order model. It is a more rigor approach to the modelling of Stirling engines which accounts for various losses that occur during engine operation. The second order analysis of the Stirling engine proved to be a more complex approach to the study of the engine’s performance. In order to enhance the first order method analysis, the second order

modelling provides a detailed description of the losses encountered during engine’s operation. A study was conducted by Boucher et al in [2] on a double free piston Stirling engine by developing dynamic balance equations, see Figure 2.1. The non-linear

dissipative effects of the fluid and the electromagnetic forces were considered in the model and the equations were solved in the time domain together with the linearized pressure equation in order to achieve a steady operation of the engine. These losses are usually

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encountered during the flow of working fluid, power transfer and transfer of heat in the engine during its operation [28]. In this analysis the design of the engines component is of a paramount importance. Various analyses have been carried out on the engines component design such as the displacer, piston, regenerator, cooler, heater and other necessary

significant components in [17, 29-31]. Pretescue employed the direct method for processes to analyse the irreversibility that occurs in the non-ideal situation of the Stirling engine cycle [32]. The second order approach was employed by Domingo for the solution of equations to analyse Stirling heat pumps [33]. The method which uses the direct integration of equations resulting from the first law for processes with finite speed was used in the second order approach in [34]. The method was considered to be successful due to the reasonable output results and its ability to predict operation in good agreement which real data of the engine performance. The second order modelling of the Stirling engine was developed by Timouni et al., where the lumped analysis, which took into account all the losses within the engine at the same time, was employed and used to optimize the General motors GPU-3 engine [35]. Martini conducted a second order analysis of the Stirling engine using the non-ideal regeneration to balance the energy equation from the Schmidt analysis. The heat transfer between the engine and the heat exchangers were taken into account. A number of investigations have been performed on the effect of irreversibility and heat losses on the performance of free piston Stirling engines. It was observed that in all the technological parameters of the Stirling engine, non-ideal regeneration and dead volume have the greatest influence on the engine performance [19, 36]. According to Popescu et al. the non-adiabatic regenerator has the most effect on the performance reduction of the engine [37]. A research was conducted by Kongtragool on the efficiency of regenerator and dead volumes on the total work done and engine efficiency without taking into consideration the heat transfer within the heat exchangers, heat source and sink [38]. The first non-isothermal analysis of the Stirling engine was carried out by

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Finkelstain. The model he developed was able to derive the finite heat transfer in the working space of the engine by the use of the heat transfer coefficient. Also the difference in the working gas temperature as it moves through the working space was calculated [39]. A general model for the simulation of Stirling engines was developed by Schulz and

Schwendig [40]. Kongtragool and Wongwise developed a thermodynamic analysis of the Stirling engine by taking into consideration the dead volume of the working space, including those in the expansion space, heater, regenerator, cooler and compression space [38]. Karabulut simulated the operation of the Stirling engine, where he carried out

kinematic and nodal thermodynamic analysis on the instantaneous temperatures of the working gas through the working space. For this study a beta type Stirling engine with a lever driven mechanism on its displacer, as shown in Figure 2.3, was used [41]. A solar Stirling engine was investigated by Mahkamov where the working process of the engine was analysed by the axisymmetric computational fluid dynamic approach [42]. A detailed analysis of the operational characteristics of a 10kW dish/Stirling system was presented by Reinalter et al [43].

The compressible flow of the working gas through the working space and the oscillations of the piston and displacer using a control-volume-based modelling were analysed by Anderson et al [44]. Schmidt presented a second order approach to the analysis of the ideal cycle of the Stirling engine. In the analysis he assumed the sinusoidal variation of the compression and expansion spaces. It was noted that the proposed analysis could not account for the non-isothermal effects and internal irreversibility caused by difference in pressure and friction of the working fluid in the working spaces of the engine [22]. A numerical study was carried out on regenerator matrix design to improve the efficiency of a Stirling engine. Numerical model of the engine was developed and a new regenerator

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design was optimized by dividing the regenerator matrix into three sections. The new design improved the overall temperature oscillations of the matrix, reduced the inflow effect on the matrix oscillations thereby improving the efficiency of the engine [45]. An investigation was conducted by Kuosa et al [31] on the impact of heat exchangers fouling on the optimum performance of the Stirling engine for the combined heat and power application. A steady state operation was achieved in the developed model considering the fouling factors of the heat exchangers. The developed model was used to evaluate and predict the fouling factors from the output performance of the engine. Formosa [10] conducted a study on the operation of a miniaturized membrane Stirling engine. The steady state operation of the engine was achieved by predicting the starting conditions of the engine and the instability problems with the aid of the stability analysis. The method of approach to obtain a simplified system without exempting the dynamics of the original systems was done by reducing the number of equations. An investigation was carried out on the development of a numerical model for a beta type Stirling engine with the rhombic drive mechanism.

This was done by considering the regenerator effectiveness, non-isothermal effects and thermal resistance of heater. The energy equations of the control volumes in the expansion and compression chambers and the regenerative channel were derived and solved. The output of the engine was improved by adjusting certain parameters such as regenerative gap, offset distance from the crank to the center or gear and the temperature of the heat source [46]. There are three stages of Stirling engine analysis deployed. The isothermal Schmidt analysis is the first stage, the second order analysis which is adiabatic is deployed at the second stage. Finally, the third order analysis was utilised for calculations [19, 39, 47]. The research conducted by Wu et al. shows the effect of the regeneration, imperfect regeneration and heat transfer on the Stirling engine cycle performance [36]. A new

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theoretical method was developed for the evaluation and improvement of the actual Stirling engine performance. Riofiro et al. developed a dynamic model of the free-piston Stirling, shown in Figure 2.4, by conducting linear analysis for the dynamic modelling elements which were employed in place of the equation of state and the output result generated was compared to the output from the experimental results [48]. Investigation was carried out on the FPSE by applying scaling effects which allows for miniaturization, as shown in Figure 2.5.

Major losses such as the heat conduction loss, pressure drop and regenerator reheat loss were defined as a function of the operating and geometrical parameters [49].

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Figure 2.4: Sunpower B-10B free-piston Stirling engine demonstrator [48].

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2.2.4 Third order modelling

The third order model is similar to the second order model but it is a more advanced approach to the modelling of Stirling engines. A study was performed on the conventional Stirling engine cycle performance which took into account the effects of incomplete heat regeneration, heat transfers within the heat exchangers and the cycle irreversibility, friction between the displacer and piston with the walls of the engine and pressure losses by Costea et al [50]. A very careful measure is required to understand the working principles of the engine in order to determine the design parameters, losses within the engine and its performance output.

The investigations carried out by Kaushik and Wu exhibited the key factors affecting the performance of the Stirling engine such as the heat transfer within the engine, the regenerator effectiveness and the rate at which the regenerator absorbs and releases the heat during the heat transfer back and forth within the heat exchangers [26, 36]. A study was conducted on a split-Stirling cryocooler with the aid of dynamic simulation by Cun-quan et al. The losses encountered such as shuttle loss, pump pressure loss and losses due to regenerator inefficiency were integrated into the mathematical model. Hence a conclusion was achieved that accounting for such losses improves agreement between the experimental and mathematical model results [51]. A beta-type Stirling engine functioning at atmospheric pressure was produced by Cinar et al and the observation on the parametric check showed that increase in the hot source temperature also increased the speed, torque and output power [52]. Some losses were analysed by Walker in the research he carried out on Stirling engines: shuttle, load and spring hysteresis losses. Due to the difficulty in determining such losses these are not usually considered in detail in the literature [13]. A quasi-stationary and adiabatic model was developed by Urieli and Berchowitz where pressure drops in the heat exchangers were considered. The calculations demonstrated that

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though there was difference between the experimental results and the model, agreement was still better than with other models. Hence the performance of the Stirling engine depends on the physical and geometrical properties, the working fluid, pressure drops, heat losses, hysteresis losses and temperature changes. Parlak et al conducted an investigation on a gamma type Stirling engine considering the thermodynamic analysis of its quasi steady flow model. The analysis was performed on the five component parts of the engine namely expansion and compression spaces and heat exchangers (heater, regenerator and cooler). The output results generated from the developed model predicted more accurate results than the models available in literature [53].

Also detailed research was carried out on Stirling engines in order to increase their performance output and analyse their operations.

A numerical simulation model was developed considering the thermal losses to optimize the engine performance. The experimental data from the GPU-3 Stirling engine was used and the results obtained showed a good correlation with prototype of the engines output. Also the influence of the physical and geometrical parameters on the engine performance was investigated in [54]. A research was performed on the design of a low temperature differential double acting Stirling engine for solar application. The engine design was determined based on Schmidt analysis and third order analysis was used to establish a complete analytical model during the design optimization stage [55].

2.2.5 Optimisation of Stirling engines

Optimisation research is being performed on improvement of designs of the cold-end and hot-end heat exchangers and regenerator of the Stirling engine based on the thermodynamic and other types of mathematical models. For example, the mathematical

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model of the Stirling engine, based on the finite dimension thermodynamics process, was used for optimisation calculations. The engine under steady state condition was analysed using the energy balance equations. Certain parameters such as the thermal efficiency and output power, were considered as the objective function in order to carry out the optimization in [56]. A one dimensional model was produced by Boer [57] for analysing Stirling engine regenerators. This model was developed from the simplified equations considering the thermal and viscous losses.

The output from the optimisation provided the optimal values of the piston phase angle and parameters of the regenerator to achieve maximum power output. A model for optimisation was developed by Senft using the principle of the forced work integrated to the classical Schmidt theory. The results obtained from the optimisation produced maximum brake work considering the optimal values generated for the swept volume ratio and phase angle [58]. Cullen et al. carried out optimisation of the geometry of the gamma-type Stirling engine. To obtain preliminary results of the modelling of an Otto/Stirling cycle hybrid engine, Schmidt model was considered. The results derived from the simulation, using direct method, were compared with the simulations with respect to the engine speed using the Schmidt model [59]. A study was performed by Orunov et al [60] on the design of the tri-generation power unit based on the alpha Stirling cycle for the production of CCHP-combined cooling, heating and power production. At the first stage of the design process, the first order model of the Stirling cycle was developed and hydraulic losses were considered. The second order model which was used after the first order took into account the hydraulic losses in the heat exchangers in order to analyse the process of the working fluid. The optimisation was performed considering dimensionless parameters such as the length of the heat exchangers (heater, regenerator and cooler) as the criteria for the procedure. The Optimisation of 20 kW Stirling engine was carried out by Zarinchang and

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Yarmahmoudi. The second order model of the engine was used for optimisation by coupling to the code named OPTIMUM. The sensitivity analysis was used to redesign the heater, cooler and regenerator. A third order program called STRENG was used to recalculate the results obtained from the optimisation with the second order model for more accurate output result prediction [61]. An investigation was conducted by Blank and Wu [62] on an extra-terrestrial Stirling engine using the irreversible thermodynamic cycle analysis powered by a solar heat source.

The optimal values of the power output and overall efficiency were determined using the finite time approach. An analysis was conducted by Erbay and Yavuz on the performance of the Stirling heat engine at maximum operating conditions. The model of the thermodynamic cycle of a Stirling engine with a regenerative heat exchanger was used to determine the power output and efficiency. The engine performance was analysed using the maximum power density technique [16]. The operating temperature for a solar powered Stirling engine was optimised with the use of mathematical model based on Stirling engine thermodynamic cycle by Costea et al [63]. The mechanical losses and fluid friction was considered using empirical correlations in the model. Lagrangian undetermined multiplier method was employed to solve the system of nonlinear equations to obtain output results.

2.3 Free piston Stirling engine modelling

The layout of the free piston Stirling engine that was invented by William T. Beale in the early 1960s resulted to be one of the assuring discoveries in the applications of the Stirling cycle [64]. It consists of a simple mechanism where direct translational vibrations of the piston and displacer produces power output [13]. Different types of heat source such as radioisotope energy, fossil fuels, solar energy, geothermal energy, coal, wood etc. can be used as the source of inut energy. The free piston Stirling engine developed by Sunpower Inc. is shown in Figure 2.6.

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Figure 2.6: Cut away view of RE-1000 free-piston Stirling engine [65].

To perform a thermodynamic analysis on the FPSE the phase angle, expansion and compression swept/clearance volumes need to be determined. In classical Stirling engine these parameters are predetermined by mechanical kinematics. Parameters such as mean pressure, external temperatures and frequency are also determined. The expansion and compression temperatures are defined by the characteristics of the heat exchangers and the properties of the working fluid.

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2.3.1 First order model

A linearized model was produced by Urieli and Berchowitz in order to analyse and predict the stable operation of the engine [19]. The Linear dynamic analysis of the FPSE produces estimated output which can be used to predict the engine design and performance [66]. The analysis carried out by Redlich and Berchowitz was to linearize the motion equation for the displacer and piston caused by the pressure of the working fluid and the gas springs [67]. The analysis predicted the algebraic relations between the thermodynamic and geometric engine parameters. It can be used to predict the performance stable operation of the engine. Organ carried out an isothermal assumption where he proposed a method to examine the mass flow rate in the engine. The mass flow rates are linked to the piston and displacer strokes, phase angle, operating frequency and temperature [30].

2.3.2 Second order model

The Hopf bifurcation method was used for the non-linear analysis of the FPSE and for its simulation deploying Schmidt and Nodal analysis. Multiple scale method was used on the beta FPSE with cube damping in the power piston chamber to examine the nonlinear system in order to create limited cycle motions. The comparison was carried out between the numerical and analytical results which gave a close prediction of the output generated [68] . Parametric check was also carried out to evaluate the different parameters that could result in instability in the Hopf results.

The thermodynamic model of the engine can be developed for the given sink and heater temperature [69]. Realistic results of the FPSE can be produced with the use of dynamic-thermodynamic studies. Thus, Kim et al carried out a study on the dynamic-thermodynamic performance of a 35W, 80W and 1.1 W solar powered free piston Stirling engines, see

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Figures 2.7 and 2.8. The results were presented for the output power over a wide range of temperatures. The engines attained an overall efficiency greater than 55% of Carnot efficiency at their design specifications [70]. A study was conducted by Formosa and Despesse on the free piston Stirling engine configuration by developing an analytical thermodynamic model. In order to achieve a more realistic output result, heat losses and irreversibility during engine operation were taken into consideration. The effectiveness of the heat exchangers, especially the regenerator, was investigated. Optimization was carried out by reducing the losses and improving engine performance [69]. A research was performed on the thermal efficiency of a low mass 35W free piston Stirling engine design. The predictions of the engines output performance in terms of power output and efficiency were similar to that of the Sunpower EG-1000 engine [71]. The study was carried out on the Free piston Stirling engine pumps by Beachely and Fronczak [72]. Another method was deployed by McGee et al. to evaluate the hydraulic power supply by introducing the mono propeller to drive the free piston hydraulic pump [73].

Considerable emissions and high noise occur due to the short combustion of the internal combustion engines.

An analysis was performed on the stable operation of the Stirling engine by Rogdakis et al. where a linear analytical model was developed. The schematic diagram of the beta free piston Stirling engine employed is shown in Figure 2.2 [9]. Linear coefficients were introduced to model the dynamics of the Stirling engine by De Monte and Benvenuto which was equivalent to the nonlinear model [12, 74]. Analysis was carried out using the linearized model of the engine with the nonlinear load by Ulusoy and Mc Caughan in [23].

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2.3.3 Third order model

Benvenuto et al proposed a method of analysing the FPSE which gave a good prediction of the stationary oscillations of the displacer and piston taking into consideration the pressure loss within the heat exchangers that in turn produced a steady amplitude of oscillation but the analysis has not been experimentally validated [75]. It also gives room to investigate the engine behaviour under various loading conditions such as change in the parameters of the spring stiffness, instantaneous pressure, damping coefficient and weight of the moving elements [76]. Therefore, an accurate global method using a thermodynamic isothermal model in agreement with the dynamic analysis is required to design a free piston Stirling engine. There is more complexity in the numerical calculations associated with the third order modelling in comparison to the second order. To model an engine in the third order, there is need to form and produce the partial differential equations of the momentum, , mass and energy conservation for a number of gas control volumes in the circuit of engine. A Stirling engine analysis code called H-FAST which employs the harmonic analysis method was developed by Huang [77] . The energy, momentum and mass conservation equations were used in a simplified manner. To obtain the correct output results, the various losses were determined. The software called PROSA developed by Thomas was described by Anderson in [78]. A one- dimensional modelling approach for a Stirling engine was produced by Anderson et al [79] which takes into consideration the compressibility in the unsteady gas flow. The empirical correlations were used to determine the losses due to the finite temperature heat transfer and flow friction. The MusSIM software was used for the modelling approach on the SM5 Stirling engine. The results obtained from the simulation showed a good agreement when compared to experimental data. For accurate prediction of power output and efficiency there is need for correct empirical correlations to calculate friction and heat transfer in the regenerator and

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heat transfer in the displacer clearance space. In addition shooting method was employed for the optimisation carried out in this study.

2.4 Optimization of free piston Stirling engines

Certain parameters such as the maximum thermal efficiency, maximum power, minimum entropy generation and maximum exergy efficiency were considered as the objective functions in order to carry out the optimization of the FPSE in [80]. A one dimensional model was produced by Boer [57] for analysing Stirling engine regenerators. The model was developed considering the thermal and viscous losses.

The output from the optimisation calculations provided the optimal values of the piston phase angle and parameters of the regenerator to achieve the maximum output power. A model for optimisation was developed by Senft using the principle of the forced work integrated to the classical Schmidt theory. The design parameters obtained from the optimisation ensured the maximum brake work at the optimal values of the swept volume ratio and phase angle [81]. The optimization of the heat exchangers of the FPSE was performed by Shoureshi on the basis of ratio of the operating temperatures, Mach number and percentage of heat exchanger dead volume [82].

A method was proposed in 1990 by Benevenuto et al for the optimization of the free piston Stirling engine in order to predict design parameters of the engine for space applications. A special consideration was used for the effects of losses due to gas hysteresis in the gas spring space with the temperature variation effect in the working space. The analysis performed resulted in the method to ascertain the dynamic behaviour of the FPSE model [74]. The free piston Stirling engine is one of the forms of the Stirling engines designed by Beale in the 60s [83]. The basic mechanical structure of the FPSE with no lateral loads

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reducing wear and providing longer running time, is the major advantage it has over other classical Stirling engines and internal combustion engines [84, 85]. To optimize a FPSE is a difficult and challenging task. The piston and displacer are driven by the spring pressure and working gas. The motions of the displacer and piston alter the volume of the working space as well as the pressure and pressure losses through the heat exchangers. Before the dynamic model of the FPSE is developed the thermodynamic properties of the engine must be defined. In order to obtain the performance of the FPSE, linearization methods were employed and behaviour of the engine was determined. With this type of analysis the steady state operation of the engine cannot be predicted accurately [86]. The evaluation of the pressure drop during the engine operation is very important for accurate prediction of operation of FPSEs. . The performance of a free piston Stirling engine, powered by the heat from an incinerator, was investigated by Hsu et al [87]. The cycle-averaged heat transfer coefficient’s value was used to determine the engine thermal efficiency and the power output for various heat source and sink temperatures. It was discovered that the optimal power output and efficiency were proportional to the temperature of the heat source. The feasibility of power production using the waste heat of an incinerator and a FPSE was investigates by Hsieh et al. The heat transfer with irreversible processes with the Lagrange multiplier method was used to optimise the engine’s output power. For this, the FPSE performance was predicted and maximum power was considered as an objective function in the procedure for optimisation [88].

The review of the optimisation methods used for designing engines shows that there has been very limited usage of Genetic algorithm methods.

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2.5 Optimization using the Genetic Algorithm procedure

The use of Genetic Algorithm has been applied previously in many engineering fields, but it has not been much used in the design of free piston Stirling engines. A brief review on application of Genetic Algorithm method for designs of engines and thermal machines is discussed in this section and the detailed description of Genetic Algorithm and its procedure in application to the FPSE is provided in Chapter 7. GA was used to optimise four parameters of the turbofan engine, such as the compression pressure ratio, bypass ratio, fan pressure ratio and Mach number. The results generated ensured the best values of the overall efficiency and thrust per mass flow rate and this case demonstrated that GA is a good tool for improving the engine performance [89]. The GA method was employed for the optimisation of a natural gas engine by Kesgin in [90]. The objective functions for optimisation were the efficiency and NOx emissions.

Also the heat exchanger, which is a very essential component of the thermal system of a heat engine, was also optimised using the GA method. A computer program for obtaining the optimal design of heat exchangers was developed by Tayal et al [91]. For the GA optimisation procedure of the ST-5 engine, HTRI program was used to determine the heat transfer area. Different strategies of GA were employed to obtain the overall minimum costs. The simulated annealing (SA) and GA methods were compared and discussed. GA optimisation method was used by Mohagheghe and Shayegan to determine the thermodynamic optimal design of heat exchangers in the heat recovery steam generator (HRSG) of the combined cycle gas turbine (CCGT). Taking into account the non-linearity of the system, the GA turned out to be the best optimisation tool [92]. The optimal design of shell and tube-heat exchangers was determines using GA method by Ponce-Ortega et al [93]. The heat exchanges were designed using the Bell-Delaware method but due to high level of the system nonlinearity, GA was employed for the optimisation. The objective